Related papers: Deciding lower-boundedness of polynomials
We introduce different notions of polynomial convexity with bounds on degrees of polynomials in $\mathbb C^n$. We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree…
The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…
We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.
In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.
In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such…
This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…
An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
This article derives lower bounds on the supremal (strict) p-negative type of finite metric spaces using purely elementary techniques. The bounds depend only on the cardinality and the (scaled) diameter of the underlying finite metric…
First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the…
We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…
We find a new lower bound for the maximal number of zeros to harmonic polynomials, $p(z)+\overline{q(z)}$, when ${\rm deg}\, p = n$ and ${\rm deg}\, q = n-2$.
We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…
The polynomials of degree $\frac{p-1}{2}$ of range sum $p$ was determined in {\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and…
In this work, we are dealing with some properties relating the zeros of a polynomial and its Mahler measure. We provide estimates on the number of real zeros of a polynomial, lower bounds on the distance between the zeros of a polynomial…
We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in $\mathbb{R}^2$.
We use the implicitization procedure to generate polynomial equality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network with hidden variables. We show how we may reduce…
This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as…
We give an example of a polynomial of degree 4 in 5 variables that is the sum of squares of 8 polynomials and cannot be decomposed as the sum of 7 squares. This improves the current existing lower bound of 7 polynomials for the Pythagoras…
We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation…